Ni--Ga bilayers are a versatile platform for exploring the competition between strongly antagonistic ferromagnetic and superconducting phases. We characterize the impact of this competition on the transport properties of highly-ballistic Al/Al2O3(/EuS)/Ni--Ga tunnel junctions from both experimental and theoretical points of view. While the conductance spectra of junctions comprising Ni (3 nm)--Ga (60 nm) bilayers can be well understood within the framework of earlier results, which associate the emerging main conductance maxima with the junction films' superconducting gaps, thinner Ni (1.6 nm)--Ga (30 nm) bilayers entail completely different physics, and give rise to novel large-bias (when compared to the superconducting gap of the thin Al film as a reference) conductance-peak subseries that we term conductance shoulders. These conductance shoulders might attract considerable attention also in similar magnetic superconducting bilayer junctions, as we predict them to offer an experimentally well-accessible transport signature of superconducting triplet pairings that are induced around the interface of the Ni--Ga bilayer. We further substantiate this claim performing complementary polarized neutron reflectometry measurements on the bilayers, from which we deduce (1) a nonuniform magnetization structure in Ga in a several nanometer-thick area around the Ni--Ga boundary and can simultaneously (2) satisfactorily fit the obtained data only considering the paramagnetic Meissner response scenario. While the latter provides independent experimental evidence of induced triplet superconductivity inside the Ni--Ga bilayer, the former might serve as the first experimental hint of its potential microscopic physical origin. Finally, we introduce a simple phenomenological toy model to confirm also from the theoretical standpoint that superconducting triplet pairings around the Ni--Ga interface can indeed lead to the experimentally observed conductance shoulders, which convinces that our claims are robust and physically justified. Arranging our work in a broader context, we expect that Ni--Ga-bilayer junctions could have a strong potential for future superconducting-spintronics applications whenever an efficient engineering of triplet-pairing superconductivity is required.